Reducing parasitic interactions in a qubit grid for surface code error correction

ABSTRACT

Methods and systems for performing a surface code error detection cycle. In one aspect, a method includes initializing and applying Hadamard gates to multiple measurement qubits; performing entangling operations on a first set of paired qubits, wherein each pair comprises a measurement qubit coupled to a neighboring data qubit in a first direction; performing entangling operations on a second set of paired qubits, wherein each pair comprises a measurement qubit coupled to a neighboring data qubit in a second or third direction, the second and third direction being perpendicular to the first direction, the second direction being opposite to the third direction; performing entangling operations on a third set of paired qubits, wherein each pair comprises a measurement qubit coupled to a neighboring data qubit in a fourth direction, the fourth direction being opposite to the first direction; applying Hadamard gates to the measurement qubits; and measuring the measurement qubits.

BACKGROUND

Large-scale quantum computers have the potential to provide fastsolutions to certain classes of difficult problems. For large-scalequantum computing to be realizable, several challenges in the design andimplementation of quantum architecture to control and program quantumhardware must be overcome. Reducing the complexity of quantumarchitecture whilst maintaining a high level of control over the quantumbits included in the quantum architecture is a crucial step in buildinga scalable quantum computer.

SUMMARY

This specification relates to reducing parasitic interactions betweenquantum bits. For example, this specification describes systems andmethods for performing a surface code error detection cycle.

In general, one innovative aspect of the subject matter described inthis specification can be embodied in methods that include the actionsof method for performing a surface code error detection cycle, themethod comprising: initializing multiple measurement qubits from asystem comprising a plurality of data qubits and a plurality ofmeasurement qubits arranged as a two-dimensional grid, wherein each dataqubit of the plurality of data qubits within the grid is coupled toneighboring measurement qubits through respective qubit couplers;applying Hadamard quantum logic gates to the initialized measurementqubits; and performing multiple entangling operations on a first set ofpaired measurement and data qubits, wherein each pair in the first setof paired measurement and data qubits comprises a measurement qubitcoupled to a neighboring data qubit in a first direction; performingmultiple entangling operations to a second set of paired measurement anddata qubits, wherein each pair in the second set of paired measurementand data qubits comprises a measurement qubit coupled to a neighboringdata qubit in a second or third direction, the second and thirddirection being perpendicular to the first direction, the seconddirection being opposite to the third direction; performing multipleentangling operations to a third set of paired measurement and dataqubits, wherein each pair in the third set of paired measurement anddata qubits comprises a measurement qubit coupled to a neighboring dataqubit in a fourth direction, the fourth direction being opposite to thefirst direction; applying Hadamard quantum logic gates to the multiplemeasurement qubits; and measuring the multiple measurement qubits todetect errors.

Other implementations of this aspect include corresponding computersystems, apparatus, and computer programs recorded on one or morecomputer storage devices, each configured to perform the actions of themethods. A system of one or more computers can be configured to performparticular operations or actions by virtue of having software, firmware,hardware, or a combination thereof installed on the system that inoperation causes or cause the system to perform the actions. One or morecomputer programs can be configured to perform particular operations oractions by virtue of including instructions that, when executed by dataprocessing apparatus, cause the apparatus to perform the actions.

The foregoing and other implementations can each optionally include oneor more of the following features, alone or in combination. In someimplementations the method further comprises applying Hadamard quantumlogic gates to data qubits in the second set of paired measurement anddata qubits that are paired with measurement qubits in the seconddirection.

In some implementations the method further comprises applying Hadamardquantum logic gates to data qubits in the second set of pairedmeasurement and data qubits that are paired with measurement qubits inthe second direction; and applying Hadamard quantum logic gates to dataqubits in the second set of paired measurement and data qubits that arepaired with measurement qubits in the third direction.

In some implementations the method further comprises applying Hadamardquantum logic gates to data qubits in the second set of pairedmeasurement and data qubits that are paired with measurement qubits inthe third direction.

In some implementations the entangling operations comprise controlled-Zquantum logic gates.

In some implementations performing multiple entangling operations on thefirst set of paired measurement and data qubits comprises: separatingthe pairs into multiple subsets of paired qubits, the multiple subsetscomprising non overlapping and non-adjacent pairs.

In some implementations the multiple subsets comprise three subsets.

In some implementations performing multiple entangling operations to thefirst set of paired measurement and data qubits comprises performingentangling operations on pairs of qubits in each of the multiple subsetsin parallel.

In some implementations performing entangling operations on pairs ofqubits in each of the multiple subsets in parallel comprises detuningeach measurement qubit in each subset in parallel.

In some implementations performing multiple entangling operations on thesecond set of paired measurement and data qubits comprises: separatingthe pairs into multiple subsets of paired qubits, the multiple subsetscomprising overlapping and diagonally adjacent pairs.

In some implementations the multiple subsets comprise four subsets.

In some implementations performing multiple entangling operations on thesecond set of paired measurement and data qubits comprises performingentangling operations on pairs of qubits in each of the multiple subsetsin parallel.

In some implementations performing entangling operations on pairs ofqubits in each of the multiple subsets in parallel comprises detuningeach measurement qubit in the each subset in parallel.

In some implementations performing multiple entangling operations on thethird set of paired measurement and data qubits comprises:separating thepairs into multiple subsets of paired qubits, the multiple subsetscomprising non overlapping and non-adjacent pairs.

In some implementations the multiple subsets comprise three subsets.

In some implementations performing multiple entangling operations on thethird set of paired measurement and data qubits comprises performingentangling operations on pairs of qubits in each of the multiple subsetsin parallel.

In some implementations performing entangling operations on pairs ofqubits in each of the multiple subsets in parallel comprises detuningeach measurement qubit in the each subset in parallel.

In some implementations the method further comprises performing aleakage removal process on the measurement and data qubits.

In some implementations performing leakage removal comprises swappingmeasurement and data qubits.

In some implementations swapping measurement and data qubits comprisesapplying a controlled-Z plus swap quantum gate.

In some implementations the method further comprises performing asubsequent surface code error detection cycle, comprising: initializingthe multiple measurement qubits; applying Hadamard quantum logic gatesto the initialized measurement qubits; performing entangling operationson the third subset of paired data and measurement qubits in parallel;performing entangling operations on the second subset of paired data andmeasurement qubits in parallel; performing entangling operations on thefirst subset of paired data and measurement qubits in parallel; applyingHadamard quantum logic gates to the multiple measurement qubits; andmeasuring the multiple measurement qubits to detect errors.

In some implementations the plurality of data qubits comprise Xmonqubits.

In some implementations the plurality of measurement qubits compriseXmon qubits.

The subject matter described in this specification can be implemented inparticular embodiments so as to realize one or more of the followingadvantages.

A quantum computing system implementing methods for reducing parasiticinteractions between qubits, as described in this specification, canperform quantum computational operations whilst reducing parasiticinteractions between qubits and introducing minimal error. The methodsdescribed in this specification can improve the robustness of thequantum computing system and improve the accuracy of computationsperformed by the quantum computing system.

Methods for reducing parasitic interactions between qubits, as describedin this specification, are scalable and allow for lenient, practicalrequirements on the physical quantum computing hardware needed toimplement the methods and perform quantum computations. For example, themethods and systems described in this specification can be implementedusing qubit frequency control architecture.

Furthermore, methods for reducing parasitic interactions between qubits,as described in this specification, can increase the efficiency ofcomputations performed by a quantum computing system implementing themethods. For example, the methods enable some quantum logic gates to beimplemented simultaneously, thus reducing the time needed to perform analgorithm.

In addition, methods for reducing parasitic interactions between qubitsin a quantum computing system, as described in this specification, arestrengthened by placing two echo pulses (rotations around the X and/or Yaxes that are designed to reduce the susceptibility of qubits to theenvironment) on idling qubits during an entangling operation on twoother qubits, greatly simplifying the algorithmic implementation ofquantum computations performed by the quantum computing system. Forexample, with two echo pulses, a sequence that suppresses noise and hasan ideal identity unitary can be constructed. Without the ability toplace two echo pulses on idling qubits during an entangling operation ontwo other qubits, it may be necessary to modify the algorithmicimplementation to deal with the unitaries of the echo pulses thatcommute through the entangling gates representing the entanglingoperation. The systems and methods described in this specification avoidthis modification.

Furthermore, methods for reducing parasitic interactions between qubits,as described in this specification, are lenient with respect to requiredqubit detunings and thresholds for parasitic coupling strength. Thepracticality and applicability of the methods described in thisspecification are therefore far reaching.

One approach to building and operating a quantum computing device isbased on surface codes that are operated as stabilizer codes. Surfacecodes provide a practical method of identifying and handling errors in atwo dimensional array of qubits. However, standard implementations ofsurface codes, e.g., implementations different to that described in thisspecification, require a dense pattern of nearest neighbor entanglingoperations. Such dense patterns can cause parasitic couplings betweenqubits that are diagonally opposed to each other, as described herein.

A quantum computing system implementing methods for reducing parasiticinteractions between qubits, as described in this specification,performs surface code cycles using a particular configuration of pairedqubits. The configuration enables the surface code to be reliablyimplemented using a closely spaced, dense two dimensional grid ofqubits. In addition, the configuration enables the surface code to beimplemented using fewer layers of entangling operations compared toother surface code implementations.

The details of one or more implementations of the subject matter of thisspecification are set forth in the accompanying drawings and thedescription below. Other features, aspects, and advantages of thesubject matter will become apparent from the description, the drawings,and the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an example quantum computing system.

FIG. 2 is a flow diagram of an example process for operating a system ofqubits.

FIG. 3A shows example schematic data qubit frequency patterns.

FIG. 3B shows an example plot showing idling error due to parasiticinteractions versus qubit frequency divided by system nonlinearity for asingle qubit.

FIG. 4 shows example data qubit and measurement qubit frequencies.

FIG. 5 is a flow diagram of an example process for performing entanglingoperations on a two dimensional array of qubits.

FIG. 6 shows example pairings of data and measurement qubits forperforming entangling operations on a two dimensional array of qubits.

FIG. 7 is an example plot of a controlled-Z quantum gate frequencytrajectory.

FIG. 8 is an example plot of a probability of parasitic occupationtransfer versus diagonal coupling strength.

FIG. 9 shows an example quantum circuit to measure a stabilizer for asurface code detection cycle.

FIG. 10 is a flow diagram of an example process for performing a surfacecode error detection cycle.

FIG. 11 shows example uniform stabilizers for a surface code detectioncycle.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

Qubits in a quantum computing system can be arranged and operated in atwo dimensional grid using nearest neighbor interactions. However, insuch a grid, couplings between qubits that are diagonally opposed toeach other are relatively large due to their adjacency. Such couplingsare parasitic—unintended and uncontrolled. For example, if two diagonalqubits form a parasitic coupling, the qubits may effect each other in anunintended and uncontrolled manner, e.g., by causing unwantedtransitions in one or both of the qubits. Unwanted transitions in aqubit can cause the state of the qubit to flip, e.g., from onecomputational state to another, or cause transitions to higher qubitlevels outside the computational subspace. Such transitions canintroduce errors into computations performed by the qubits. Minimizingparasitic couplings between qubits when operating a quantum computer,particularly at scale, is therefore an important task in quantumcomputing.

Example Operating Environment

FIG. 1 is a block diagram of an example quantum computing system 100.The example quantum computing system 100 includes multiple qubits 102(represented as open and filled-in circles) and a qubit controllermodule 104. The example quantum computing system 100 is an example of asystem that may be used to perform quantum algorithmic operations,simulations or computations.

The multiple qubits 102 are arranged as a two dimensional grid 112. Forclarity, the two dimensional grid 112 depicted in FIG. 1 includes 7×7qubits, however in some implementations the system 100 may include asmaller or a larger number of qubits. The multiple qubits 102 interactwith each other through multiple qubit couplers, e.g., qubit coupler114. The multiple qubit couplers define nearest neighbor interactionsbetween the multiple qubits 102. In some implementations, the strengthsg of the multiple qubit couplers are tunable parameters. In some cases,the multiple qubit couplers included in the quantum computing system 100may be couplers with a fixed coupling strength.

In some implementations the multiple qubits 102 may include data qubits,e.g., the open circles such as qubit 118, and measurement qubits, e.g.,the filled in circles, such as qubit 116. A data qubit is a qubit thatparticipates in a computation being performed by the system 100. Ameasurement qubit is a qubit that may be used to determine an outcome ofa computation performed by the data qubit. That is, during a computationan unknown state of the data qubit is transferred to the measurementqubit using a suitable physical operation and measured via a suitablemeasurement operation performed on the measurement qubit.

The multiple data qubits are each configured to operate at qubitfrequencies from a respective frequency region. For example, each dataqubit may be configured to operate at a respective data qubit frequencyfrom one of multiple data qubit frequency regions. Similarly, eachmeasurement qubit may be configured to operate at a respectivemeasurement qubit frequency from one of multiple measurement qubitfrequency regions.

The qubit frequency regions may include computational qubit frequencyregions. For example, each data or measurement qubit may be configuredto operate at a qubit frequency from a respective computational qubitfrequency region, e.g., when the qubit is involved in a computation oralgorithmic operation. A qubit may be said to operate at a qubitfrequency from a computational qubit frequency region if transitionsbetween qubit computational basis states occur at the qubit frequency.

The qubit frequency regions may include idling qubit frequency regions.For example, each data or measurement qubit may be configured to operateat a qubit frequency from a respective idling qubit frequency region,e.g., when the qubit is idling and not involved in an interaction orcomputational operation. A qubit may be said to operate at a qubitfrequency from an idling qubit frequency region for a set duration oftime if an intended unitary for the qubit is unity II for the durationof time. The qubit is therefore not engaging in entangling quantum logicgate operations—only echo pulses with unity unitary are applied to thequbit, where an echo pulse is defined a rotation around the X and/or Yaxis that is designed to reduce the susceptibility of qubits to theenvironment. In other words, the qubit may be said to be resting at afrequency from the idling qubit frequency region that minimizesinteractions between the qubit and other qubits, whilst other qubitsperform interactions or undergo unitary operations.

The qubit frequency regions may include interaction qubit frequencyregions. For example, a data and measurement qubit may be configured tooperate at respective qubit frequencies from a respective interactionfrequency region, e.g., when the data and measurement qubit interact.

The qubit frequency regions may include a readout and reset frequencyregion. For example, a measurement qubit may be configured to operate ata qubit frequency from a respective readout and reset frequency region,e.g., when a measurement operation is being performed on the measurementqubit. A qubit may be said to operate at a qubit frequency from areadout and reset qubit frequency region if the qubit frequency is nearor in alignment with the operating frequency of a readout resonator orother measurement device to bring about a measurement or resetoperation.

The data qubits and measurement qubits are arranged such that the qubitcouplers define nearest neighbor interactions between data qubits andmeasurement qubits. That is, each data qubit is coupled to multiplemeasurement qubits, and each measurement qubit is coupled to multipledata qubits.

In other implementations, the multiple qubits 102 may not be separatedinto multiple data qubits and multiple measurement qubits. For example,in cases where the system 100 includes a quantum computer that is notimplementing the surface code, e.g., a quantum computer that is used toperform one or more quantum algorithms such as supremacy algorithms, thesystem may not need to distinguish between data and measurement qubits.In these examples, the disclosed systems and methods can be used toreduce the number of layers of quantum logic gates required to performthe quantum algorithms, therefore speeding up the algorithm and reducingthe error of the algorithm.

In cases where there is no distinction between data qubits andmeasurement qubits, the same processes described herein can be used. Forexample, the qubits may be configured to operate at qubit frequencieswithin respective frequency regions. Some qubits may be configured tooperate at respective qubit frequencies within one of multiple differentfirst qubit frequency regions. Other qubits may be configured to operateat respective qubit frequencies within one of multiple different secondqubit frequency regions. The qubit frequency regions in the multiplefirst and second qubit frequency regions may include respectivecomputational frequency regions, idling frequency regions, interactionfrequency regions and readout/reset frequency regions.

As described herein, the example two dimensional grid 112 may includeparasitic couplings between qubits that are diagonally opposed to eachother, e.g., parasitic coupling 120. In some cases, parasitic couplingsbetween qubits include a nonzero parasitic coupling strength g_(diag).For example, in some cases the parasitic coupling strength may takevalues of up to g_(diag)/π˜5 MHz.

The multiple qubits 102 in two dimensional grid 112 are operated via thequbit controller module 104. The qubit controller module 104 may operatethe qubits 102 by controlling the frequencies of the qubits 102, e.g.,according to qubit operating frequencies 108. The qubit operatingfrequencies are dependent on the type of qubits included in the system100 and on the operations being performed by the system. Example qubitoperating frequency patterns for reducing parasitic interactions betweendiagonal qubits are described in detail below with reference to FIGS. 2to 4.

For example, the qubit controller module 104 may control individualfrequencies of the qubits 102 such that the frequency of one or more ofthe qubits are adjusted towards or away from a frequency, e.g., anentangling operation frequency 106, of an excitation pulse generated byan excitation pulse generator 110 on an excitation driveline 124.Excitation pulses generated by the excitation pulse generator 110 mayinclude pulses whose frequencies implement quantum operations, e.g.,quantum logic gates. For example, the excitation pulse generator 110 maybe configured to generate excitation pulses whose frequencies cause oneor more qubits to perform entangling operations, e.g., controlled-Zgates. Performing entangling operations on a two dimensional grid ofqubits is described in more detail below with reference to FIGS. 2 to 5.

The multiple qubits may be coupled to the excitation driveline viarespective couplers, e.g., coupler 126. In some cases the couplers maybe capacitive couplers, e.g., realized by a microwave line runningadjacent to a qubit capacitor. For convenience, a global excitationdriveline is illustrated in FIG. 1. However, in some implementations thesystem 100 may include multiple excitation drivelines, e.g.,corresponding to multiple qubits.

The qubit controller module 104 may be configured to tune thefrequencies of the qubits 102 through one or more qubit frequencycontrol lines, e.g., qubit frequency control line 122. For convenience,one qubit frequency control line is shown in FIG. 1. However, in someimplementations the system 100 may include multiple qubit frequencycontrol lines, e.g., corresponding to each of the multiple qubits 102.The qubit frequency control lines may be supplied by in-plane wiring orout-plane wiring.

The type of qubit controller 112 that the system 100 utilizes isdependent on the type of qubits the system uses. As an example, qubitsthat are realized via atomic, molecular or solid-state quantum systemstypically have energy separation of the relevant qubit levels in themicrowave or optical domain. The states of such qubits may bemanipulated and controlled using external fields, such as microwave oroptical fields. In such cases, as an example, mode-locked lasers mayserve as qubit controllers due to their broad-band optical spectra thatfeature both radio frequency and microwave structure. In anotherexample, the qubit controller could include a collection of individualqubit controllers realized by a radio frequency generator as well as oneor a collection of global excitation controllers realized by a radiofrequency or microwave generator. In both cases, the qubit controllermay be operated manually or connected to a computer and controlled viasuitable software allowing for specifying and automatically running therequired qubit operations.

Programming the Hardware: Quantum Bit Frequency Patterns for DecreasingParasitic Interactions

For convenience, the disclosure provided below with reference to FIGS. 2to 4 is described with reference to a system of multiple qubits thatinclude multiple data qubits and multiple measurement qubits thatinteract via qubit couplers defining nearest neighbor interactionsbetween data and measurement qubits. However, this is one example of asystem of qubits that may be programmed and operated using thetechniques described herein. For example, in some implementations thefollowing techniques may be used to program and operate a system ofqubits that do not distinguish between data or measurement qubits. Forexample, in cases where the system of qubits is used to perform quantumalgorithms, e.g., supremacy algorithms, the qubits may beindistinguishable. In settings where it is not required to distinguishbetween data and measurement qubits, the following arrangements andprocesses may still be used.

FIG. 2 is a flow diagram of an example process 200 for operating asystem of qubits. For convenience, the process 200 will be described asbeing performed by a system of one or more quantum or classicalcomputers located in one or more locations. For example, the process 200can be implemented using the qubit controller 104 of system 100described above with reference to FIG. 1.

The system of qubits includes qubits that interact via qubit couplersdefining nearest neighbor interactions. In some implementations thesystem of qubits may include multiple data qubits and multiplemeasurement qubits that interact via qubit couplers defining nearestneighbor interactions between data and measurement qubits. The system ofqubits is arranged as a two dimensional grid and each data qubit of themultiple data qubits is coupled to multiple measurement qubits throughmultiple qubit couplers. Each data qubit is configured to operate at adata qubit frequency from one of multiple data qubit frequency regions.Each measurement qubit is configured to operate at a measurement qubitfrequency from one of multiple measurement qubit frequency regions.Example data qubit frequency regions and measurement qubit frequencyregions are described below with reference to steps 202 and 204. Anexample system of qubits is illustrated with reference to FIG. 1.

The system operates a first data qubit from the multiple data qubits ata first data qubit frequency from a first data qubit frequency region(step 202). For example, as illustrated in schematic data qubitfrequency pattern 300 described below with reference to FIG. 3A, dataqubit 302 may be operated at a data qubit frequency b from a respectivedata qubit frequency region B.

The system operates a second data qubit from the multiple data qubits ata second data qubit frequency from a second data qubit frequency region(step 204). The second data qubit is a data qubit that is diagonal tothe first data qubit in the two-dimensional grid. For example, asillustrated in schematic data qubit frequency pattern 300 describedbelow with reference to FIG. 3A, the system may operate a second qubit,e.g., qubit 304, at a data qubit frequency a from a respective dataqubit frequency region A.

The second data qubit frequency and the second data qubit frequencyregion is different to the first data qubit frequency and the first dataqubit frequency region, respectively. For example, as illustrated inschematic data qubit frequency pattern 300 described below withreference to FIG. 3A, first data qubit 302 may be operated at a dataqubit frequency b from a respective data qubit frequency region B, andsecond data qubit 304 that is diagonal to data qubit 302, may beoperated at data qubit frequency a from a different data qubit frequencyregion A.

In some implementations the system may further operate third data qubitfrom the multiple data qubits at the second data qubit frequency. Thethird data qubit is different to the second data qubit and is diagonalto the first data qubit in the two-dimensional. For example, asillustrated in schematic data qubit frequency pattern 300 describedbelow with reference to FIG. 3A, the system may operate a first dataqubit 302 at a data qubit frequency b from a respective data qubitfrequency region B, a second data qubit 304 that is diagonal to dataqubit 302 at data qubit frequency a from a different data qubitfrequency region A, and a third data qubit 306 that is different tosecond data qubit 304 and is diagonal to the first data qubit 302.

In some implementations the system may further operate a fourth dataqubit and a fifth data qubit from the multiple data qubits at arespective fourth data qubit frequency from a third data qubit frequencyregion and fifth data qubit frequency region from a third data qubitfrequency region. The third data qubit frequency region is different tothe first data qubit frequency region. The fourth and fifth data qubitsare diagonal to the first data qubit.

For example, as illustrated in schematic data qubit frequency pattern300 described below with reference to FIG. 3A, the system may operate afirst data qubit 302 at a data qubit frequency b from a respective dataqubit frequency region B, a second data qubit 304 at data qubitfrequency a from a data qubit frequency region A, a third data qubit 306at a data qubit frequency a′ from the data qubit frequency region A, afourth data qubit 310 at a data qubit frequency a from data qubitfrequency region A, and a fifth data qubit 308 at a data qubit frequencya′ from data qubit frequency region A. In some implementations the thirddata qubit frequency region may be the same as the second data qubitfrequency region. In other implementations the third data qubitfrequency region may be different to the second data qubit frequencyregion, e.g., data qubits 308 and 310 may operate at data qubitfrequencies c′ and c, respectively.

As illustrated in schematic data qubit frequency pattern 300 of FIG. 3A,in some implementations the multiple data qubit frequency regionsinclude two data qubit frequency regions, e.g., a first region A and asecond region B. In these implementations, data qubit frequencies fromeach region may be offset by −2η, where η represents nonlinearity of thesystem. For example, data qubit frequencies aϵA and bϵB may differ by2η. One explanation of why the frequency differences depend on thenonlinearity of the system is as follows. An idling qubit may beaffected by parasitic interactions from diagonal qubits. To minimizethese effects, a frequency difference between diagonal qubits isoptimally selected. The influence of diagonal qubits can be expressed bythe nonlinearity of the system. Therefore, by analyzing the nonlinearityof the system a range of frequency values that locally minimizes theinfluence of diagonal qubits can be found. That is, the systemnonlinearity provides an indication of how to minimize parasiticinteractions between qubits and therefore how to minimize errors in thesystem. An example plot showing idling ZZ errors (y axis) from frequencyshifts due to parasitic interactions from diagonal qubits versus qubitfrequency divided by system nonlinearity (x axis) for a single qubit isshown in FIG. 3B.

In addition, in some implementations data qubit frequencies from aparticular data qubit frequency region may include frequencies within apredetermined frequency region, e.g., a predetermined frequency regionof width 10 MHz. For example, data qubits 304 and 306 of FIG. 3A mayoperate at data qubit frequencies a and a′, respectively, where a and a′differ by approximately 10 MHz.

Furthermore, in some implementations swapping can be avoided by ensuringthat the difference between data qubit frequencies within apredetermined frequency region is larger than a next-nearest neighborcoupling constant g. For example, the difference between data qubitfrequencies a, a′ of qubits 304 and 306 that are diagonal to qubit 302may be larger than g, e.g., (a−a′)>>g. For a next-nearest neighborcoupling constant g=1 MHz, a detuning between a and a′ of 10 MHz isacceptable.

As illustrated in schematic data qubit frequency pattern 350 of FIG. 3A,in some implementations the multiple data qubit frequency regionsinclude four data qubit frequency regions, e.g., regions A, B, C and D.Multiple frequency regions enable qubits to be “parked” and individuallycontrolled, e.g., using a global XY excitation driveline. For example,for η=200 MHz, data qubits may be parked or operated at frequenciesbetween 6 and 7 GHz, e.g., at 6.7 GHz, 6.3 GHz, 6.8 GHz, and 6.2 GHz.

In these implementations, the multiple data qubit frequency regions mayinclude a first idling frequency region. A data qubit may be configuredto operate at an idling frequency when the data qubit is not activelyinvolved in an algorithmic computation being performed by the system ofqubits and is idling.

The multiple data qubit frequency regions may further include a firstecho operation frequency region. A data qubit may be configured tooperate at an echo operation frequency when an echo operation is beingperformed on the data qubit.

The multiple data frequency regions may further include a first singlequbit gate frequency region. A data qubit may be configured to operateat a single qubit gate frequency region when a single qubit quantumgate, e.g., a Hadamard quantum logic gate or a Pauli X, Y or Z quantumlogic gate, is being performed on the data qubit.

The multiple data frequency regions may further include an interactionfrequency region. A data qubit may be configured to operate at aninteraction frequency region when the data qubit is interacting with aneighboring measurement qubit, e.g., when entangling operations areperformed on a paired data and neighboring measurement qubit. Exampledata qubit frequency regions are illustrated with reference to FIG. 4.

In implementations where the multiple data qubit frequency regionsinclude four data qubit frequency regions, operating the system ofqubits may include, for each data qubit, operating the data qubit at adata qubit frequency from a data qubit frequency region, where eachother data qubit that is diagonal to the data qubit is operated at arespective other data qubit frequency from a different data qubitfrequency region, and where opposing other data qubits that are diagonalto the data qubit are operated at respective other data qubitfrequencies from different data qubit frequency regions.

For example, as illustrated in schematic data qubit frequency pattern350 of FIG. 3A, data qubit 352 may be operated at a data qubit frequencyb from a respective data qubit frequency region B, and each other dataqubit that is diagonal to data qubit 352, e.g., data qubits 354, 356,358 and 360, may be operated at data qubit frequencies a, a′, c, or c′from different data qubit frequency regions A and C. Opposing other dataqubits that are diagonal to the data qubit 352, e.g., data qubits 356and 358, or data qubits 354 and 360, operate at respective data qubitfrequencies from different data qubit frequency regions. That is, dataqubit 356 operates at a frequency from data qubit frequency region A anddata qubit 358 operates at a frequency from different data qubitfrequency region C. Similarly, data qubit 354 operates at a frequencyfrom data qubit frequency region A and data qubit 360 operates at afrequency from different data qubit frequency region C.

In these implementations, data qubit frequencies corresponding todiagonal data qubits may be offset by −2η, e.g., (a−b)/η≥2. For example,in the schematic data qubit frequency pattern 350, the data qubitfrequencies may be set as a=0, b=−2η, c=0.5η, d=−2.5η. In someimplementations η=0.2 GHz, giving frequency ranges of approximately0.8-1.0 GHz for the frequency domain in which similar qubits—data ormeasurement qubits—can be parked. For example, if all data qubits areparked at a frequency between 6-7 GHz, e.g. the 0.8-1 GHz range, and allmeasurement qubits are parked between 4-5 GHz, interactions may occurbetween 5 and 6 GHz. However, other layouts are also possible.

In addition, in some implementations data qubit frequencies from aparticular data qubit frequency region may include frequencies within apredetermined frequency region, e.g., a predetermined frequency regionof width 10 MHz. For example, data qubits 354 and 356 of FIG. 3A mayoperate at data qubit frequencies a and a′, respectively, where a and a′differ by approximately 10 MHz.

Furthermore, in some implementations, the difference between other dataqubit frequencies that are diagonal to the qubit, e.g., data qubitfrequencies a, a′ or c, c′, may be larger than the nearest neighborcoupling constant g, e.g., (a−a′)>>g.

As described herein, e.g., with reference to FIG. 4, the above describeddata qubit frequency pattern facilitates a dense pattern on entanglingoperations, e.g., controlled-Z quantum logic gates, to be executed withno nearest neighboring—including diagonal—qubits at a same frequency.Parasitic interactions can therefore be reduced.

The data qubit properties described above can also be applied to themultiple measurement qubits in the system of qubits. For example,operating the system of qubits may further include, for each measurementqubit, operating the measurement qubit at a measurement qubit frequencyfrom a measurement qubit frequency region, where each other measurementqubit that is diagonal to the measurement qubit is operated at arespective other measurement qubit frequency from a differentmeasurement qubit frequency region.

In some implementations the multiple measurement qubit frequency regionsinclude two measurement qubit frequency regions. The measurement qubitfrequency and the other measurement qubit frequency may differ by 2η.The other measurement qubit frequencies may include frequencies within apredetermined frequency region, optionally including a predeterminedfrequency region of width 10 MHz. The difference between othermeasurement qubit frequencies within a predetermined frequency regionmay be larger than the nearest neighbor coupling constant g.

In some implementations the multiple measurement qubit frequency regionsmay include four measurement qubit frequency regions, optionallyincluding a second idling frequency region, a second echo operationfrequency region, a second single qubit gate frequency region, and asecond interaction frequency region. For example, data qubits may beparked or operated at a frequency between 6 and 7 GHz, measurementqubits may be parked or operated at a frequency between 4 and 5 GHz, andinteractions between qubits may occur between 5 and 6 GHz.

In these examples, operating the system of qubits may further include,for each measurement qubit, operating the measurement qubit at ameasurement qubit frequency from a measurement qubit frequency region,where each other measurement qubit that is diagonal to the measurementqubit is operated at a respective other measurement qubit frequency froma different measurement qubit frequency region, and where opposing othermeasurement qubits that are diagonal to the measurement qubit areoperated at respective other measurement qubit frequencies fromdifferent measurement qubit frequency regions.

In some implementations diagonal measurement qubits may be offset by 2η,optionally wherein η=0.2 GHz. Other measurement qubit frequencies from asame measurement qubit frequency region may include frequencies within apredetermined frequency region, optionally including a predeterminedfrequency region of width 10 MHz. The difference between othermeasurement qubit frequencies from a same measurement qubit frequencyregion is larger than g.

In some implementations, a readout and reset frequency region that isadjacent to one of the multiple measurement qubit frequency regions maybe included. Placing a readout and reset frequency region adjacent tothe measurement qubit frequency regions enables the frequencies of thequbits to be tuned close to readout resonators to get a large dispersionand therefore a large measurement signal. In addition, placing thereadout and reset frequency region adjacent to the measurement qubitfrequency regions enables a reset operation of the measurement qubitswith the readout resonators. Furthermore, placing the readout and resetfrequency region adjacent to the measurement qubit frequency regionsenables movement of the measurement qubits beyond the readoutresonators, allowing for movement of the data qubits close to thereadout resonators to get large dispersion and a large signal, withouthaving negative effects on the measurement qubits.

FIG. 3A shows example schematic data qubit frequency patterns 300 and350. Example schematic data qubit frequency pattern 300 shows multipledata qubits, e.g., data qubits 302, 304, 306, 308 and 310, coupled tomultiple measurement qubits via nearest neighbor interactions. Theexample schematic data qubit frequency pattern 300 shows data qubitsoperated at data qubit frequencies a, a′, b, b′ from two data qubitfrequency regions A and B. Each data qubit in the example schematic dataqubit frequency pattern 300 is operated at a data qubit frequency thatis different to the data qubit frequencies that diagonally adjacent dataqubits operate at. For example, data qubit 302 operates at data qubitfrequency b from frequency region B, whilst its diagonally adjacent dataqubits 304, 306, 308 and 310 operate at data qubit frequencies a, a′, a′and a, respectively, from frequency region A.

Example schematic data qubit frequency pattern 350 shows multiple dataqubits, e.g., data qubits 352, 354, 356, 358 and 360, coupled tomultiple measurement qubits via nearest neighbor interactions. Theexample schematic data qubit frequency pattern 350 shows data qubitsoperated at data qubit frequencies a, a′, b, b′, c, c′ and d, d′. Thedata qubit frequencies may be frequencies from four respective dataqubit frequency regions, e.g., regions A, B, C and D. Each data qubit inthe example schematic data qubit frequency pattern 350 is operated at adata qubit frequency that is different to the data qubit frequenciesthat diagonally adjacent data qubits operate at. In addition, qubitsthat are diagonally adjacent to a data qubit and diagonally opposed toeach other, e.g., qubits 356 and 358, operate at differing data qubitfrequencies. For example, data qubit 352 operates at data qubitfrequency b from frequency region B, whilst its diagonally adjacent dataqubits 354 and 360, since they are diagonally opposed to each other,operate at data qubit frequencies a and c′, respectively, from frequencyregions A and C. Similarly, its diagonally adjacent data qubits 356 and358, since they are diagonally opposed to each other, operate at dataqubit frequencies a′ and c, respectively, from frequency regions A andC.

FIG. 4 shows example data qubit and measurement qubit frequencies 400.The example data qubit and measurement qubit frequencies includes 9different frequencies ranging over frequency ranges 410, 412 and 414.Four frequencies are data qubit frequencies 402. Four frequencies aremeasurement qubit frequencies 406. One of the data qubit frequencies isan interaction frequency 404. Similarly, one of the measurement qubitfrequencies is an interaction frequency 404. One frequency is a readoutand reset frequency 408. This configuration of frequencies enables adense pattern on entangling operations, e.g., controlled-Z quantum logicgates, to be executed with no neighboring—including diagonal—qubits at asame frequency. Parasitic interactions can therefore be reduced, sincethe qubits are geometrically well separated.

The data qubit frequencies 1, 2 in the frequency range 410 are idlingdata qubit frequencies. Similarly, the measurement qubit frequencies 3,4 in the frequency range 412 are idling measurement qubit frequencies.The data qubit frequencies 3, 4 and measurement qubit frequencies 1, 2in the frequency range 414 are qubit frequencies for qubits that arebeing manipulated. In some implementations one or more frequencies infrequency ranges 410 and 412 or in frequency range 414 can also bechosen for globally applied single qubit gates.

The example data qubit and measurement qubit frequencies 400 furtherincludes an additional readout and reset frequency 416 for data qubits.The additional readout and reset frequency 416 is shown as being locatedat a higher frequency than the data qubit frequencies 402, i.e., abovethe data qubit frequencies 402 with reference to the illustration 400.The additional readout and reset frequency 416 could enable data qubitsto be read out or reset while interacting other pairs of qubits. Thiscould be beneficial in various settings. An example setting includesperforming surface code error detection in superconducting hardwaresince when excessively excited (leaked) states are removed, measurementqubits become data qubits and vice versa. In this setting measuring dataqubits whilst interacting with other pairs of qubits could be verybeneficial.

Programming the Hardware: Simultaneous Quantum Bit Detuning for ReducingParasitic Interactions

For convenience, the techniques described with reference to FIGS. 5 to 8relate to a system of multiple qubits that include multiple data qubitsand multiple measurement qubits that interact via qubit couplersdefining nearest neighbor interactions between data and measurementqubits. However, this is one example of a system of qubits that may beprogrammed and operated using the following techniques. For example, insome implementations the following techniques may be used to program andoperate a system of qubits that do not distinguish between data ormeasurement qubits. For example, in cases where the system of qubits isused to perform quantum algorithms, e.g., supremacy algorithms, thequbits may be indistinguishable. In settings where it is not required todistinguish between data and measurement qubits, the followingtechniques and arrangements may still be used.

FIG. 5 is a flow diagram of an example process 500 for performingentangling operations using a system of qubits. For convenience, theprocess 500 will be described as being performed by a system of one ormore quantum or classical computers located in one or more locations.For example, the process 500 can be implemented using the qubitcontroller module 104 of system 100 described above with reference toFIG. 1. In some implementations, the process 500 may be performed inconjunction with the frequency patterns described above with referenceto FIGS. 2 to 4.

The system of qubits includes multiple qubits and multiple qubitcouplers defining nearest neighbor interactions between the multiplequbits. In some implementations the multiple qubits may include multipledata qubits, multiple measurement qubits, and multiple qubit couplersdefining nearest neighbor interactions between the data qubits andmeasurement qubits. The system of qubits is arranged as a twodimensional grid and each data qubit of the multiple data qubits iscoupled to multiple measurement qubits through respective qubitcouplers. An example two dimensional grid is illustrated above withreference to FIG. 1.

The system pairs multiple data qubits with respective neighboringmeasurement qubits (step 502). In some implementations the system maypair the multiple data qubits and measurement qubits intonon-overlapping pairs. For example, each data qubit that is paired witha respective neighboring measurement qubit may not be paired withanother neighboring measurement qubit. Similarly, each measurement qubitthat is paired with a respective neighboring data qubit may not bepaired with another neighboring data qubit. An example pairing of dataqubits and neighboring measurement qubits into non-overlapping pairs isillustrated in the example two dimensional qubit grid 600 of FIG. 6.

Alternatively or in addition, the system may pair multiple data qubitsand respective neighboring measurement qubits into pairs with parallelqubit couplers. For example, with reference to the two dimensional grid112 shown above with reference to FIG. 1, the system may pair dataqubits with measurement qubits that are directly above or below the dataqubits. In this configuration, the paired data and measurement qubitsmay be described as having north-south parallel couplers.

In cases where the system pairs multiple data qubits with respectiveneighboring measurement qubits into non-overlapping pairs, the parallelcouplers have a same direction. That is, each measurement qubit may bepaired with a respective neighboring data qubit to its north (or itssouth). An example pairing of data qubits and neighboring measurementqubits into non-overlapping pairs with north-south parallel couplers isillustrated in the example two dimensional grid 600 of FIG. 6. In somecases, e.g., those where the system pairs multiple data qubits withrespective neighboring measurement qubits into overlapping pairs, theparallel couplers may have different directions. That is, somemeasurement qubits may be paired with a first data qubit in a northerlydirection and a second data qubit in a southerly direction. In theexample two dimensional grid 600, the qubits are coupled with couplersin a northerly direction only, as indicated by the arrow 626.

As another example, with reference to the two dimensional grid 112 shownabove with reference to FIG. 1, the system may pair data qubits withmeasurement qubits that are directly to the right or left of the dataqubits. In this configuration, the paired data and measurement qubitsmay be described as having east-west parallel couplers.

In cases where the system pairs multiple data qubits with respectiveneighboring measurement qubits into non-overlapping pairs, the parallelcouplers have a same direction. That is, each measurement qubit may bepaired with a respective neighboring data qubit to its west (or itseast). In cases where the system pairs multiple data qubits withrespective neighboring measurement qubits into overlapping pairs, theparallel couplers may have different directions. That is, somemeasurement qubits may be paired with a first data qubit in a westerlydirection and a second data qubit in an easterly direction. An examplepairing of data qubits and neighboring measurement qubits intooverlapping pairs with east-west parallel couplers of differentdirections is shown in the example two dimensional grid 650 of FIG. 6.In the example two dimensional grid 650, the qubits are coupled withcouplers in both an easterly and westerly direction, as indicated by thearrows 628 and 630.

In some implementations the system may pair a subset of the multipledata qubits with respective neighboring measurement qubits. For example,the system may pair multiple data qubits with respective neighboringmeasurement qubits such that each paired data qubit and measurementqubit is nonadjacent to other paired data qubits and measurement qubits.Example nonadjacent pairs of data qubits and respective neighboringmeasurement qubits are illustrated in the example two dimensional grid600 of FIG. 6.

In some implementations the system may pair multiple data qubits withrespective neighboring measurement qubits into multiple subsets ofpaired data and measurement qubits. For example, the system may repeatthe pairing process described above over multiple subsets, e.g., untileach qubit in the system of qubits is paired with at least one otherqubit.

In some cases, the multiple subsets of paired data and measurementqubits may include non-overlapping subsets of paired data andmeasurement qubits. For example, the example two dimensional grid 600 ofFIG. 6 illustrates multiple non-overlapping subsets of paired data andrespective neighboring measurement qubits. In the example twodimensional grid 600, each subset includes non-adjacent pairs of paireddata and respective neighboring measurement qubits. In other cases, themultiple subsets of paired data and measurement qubits may includeoverlapping subsets of paired data and measurement qubits. For example,the example two-dimensional grid 650 of FIG. 6 illustrates multipleoverlapping subsets of paired data and respective neighboringmeasurement qubits.

The system performs entangling operations on each paired data andmeasurement qubit in parallel (step 504). For example, the system mayapply a two-qubit quantum logic gate, e.g., a controlled-Z quantum logicgate, to each paired data and measurement qubit in parallel. Sincevariations in the frequency amplitudes of applied entangling operationscan occur, performing entangling operations on each paired data andmeasurement qubit in parallel is understood to mean performingentangling operations on each paired data and measurement qubit inparallel to the extent that the hardware used to perform process 500allows. Example variations are described in more detail below.

In cases where the system generates multiple subsets of paired dataqubits and respective neighboring measurement qubits, as described abovewith reference to step 504, the system may perform an entanglingoperation on each paired data and measurement qubit within a respectivesubset in parallel. To perform entangling operations on each data andmeasurement qubit in the system of qubits, the system may sequentiallyperform entangling operations on the paired data and measurement qubitswithin each subset. In some implementations the order in which thesystem selects subsets to perform entangling operations on may bearbitrary.

Due to the configuration of the paired qubits, as described above withreference to step 504, each qubit involved in the entangling operations(or each qubit involved in one sequential application of entanglingoperations on a subset of paired qubits) is either non-adjacent to otherqubits involved in the entangling operations, or has the same type onthe diagonal. For example, in cases where the data qubits andneighboring measurement qubits have been paired into non-adjacent pairsof data and neighboring measurement qubits, as illustrated in group 602of FIG. 6, qubits involved in one entangling operation are non-adjacentto qubits involved in other entangling operations. As another example,in cases where the data qubits and neighboring measurement qubits havebeen paired into overlapping subsets of paired data and measurementqubits with parallel couplers, as illustrated in group 608 of FIG. 6,qubits involved in one entangling operation have the same type on thediagonal, e.g., qubit 610 and 612.

Therefore, when the entangling operations are performed on each paireddata and measurement qubit in parallel, each measurement qubit can bedetuned without crossing the resonance of another measurement qubit thatis performing a similar frequency trajectory on its corresponding dataqubit. In fact, since the entangling operations are performed inparallel, the detuning Δf between the diagonal qubits is constant (ornear constant, see below), therefore no occupation transfer fromdiagonal interactions will arise. In addition, each data qubit canperform part of the trajectory—they do not have to remain at constantfrequency. For example, the data qubits may perform a frequencytrajectory that moves towards the measurement qubits. The advantages ofthe method performed by the system remain.

To perform the entangling operation of each paired data and measurementqubit in parallel, the system detunes each measurement qubit in thepaired data and measurement qubits in parallel. As described herein,detuning each measurement qubit in the paired data and measurementqubits in parallel may include maintaining constant, or near constant,detuning Δf between the measurement qubits in the paired data andmeasurement qubits. For example, the system may maintain detuningfrequencies from a predetermined range of frequencies, e.g., frequencieswithin a 100 MHz range such as between 500 MHz and 400 MHz or within a200 MHz range such as between 700 MHz and 500 MHz. In cases where thesystem pairs multiple data qubits with respective neighboringmeasurement qubits into multiple non overlapping subsets of paired dataand measurement qubits, the system may perform an entangling operationon each paired data and measurement qubit in the subset approximately inparallel for each of the multiple subsets.

In some implementations the system may perform an entangling operationon each paired data and measurement qubit by applying an entanglingoperation frequency trajectory to the paired data and measurementqubits. An example controlled-Z quantum gate frequency trajectory thatmay be applied to one or more paired data and respective neighboringmeasurement qubits is shown with reference to FIG. 7.

In some implementations the system may apply respective entanglingoperation frequency trajectories to different paired data andmeasurement qubits. In these implementations, variations between therespective entangling operation frequency trajectories may be maintainedbelow a predetermined threshold. Such variations can occur due to, forexample, variations in control pulse amplitudes, e.g., as omitted by theexcitation drivelines described herein with reference to FIG. 1.

FIG. 6 shows example pairings of data and measurement qubits forperforming entangling operations on a first two dimensional array ofqubits 600 and a second two dimensional array of qubits 650. Both twodimensional arrays of qubits 600 and 650 include multiple data qubits,e.g., data qubits 614 and 616, and multiple measurement qubits, e.g.,measurement qubit 618 and 620. Each data qubit of the multiple dataqubits is coupled to multiple neighboring measurement qubits throughrespective qubit couplers, as described herein with reference to FIG. 1.

Each paired data and neighboring measurement qubit in the twodimensional array 600 does not overlap with another paired data andneighboring measurement qubit. In addition, each paired data andneighboring measurement qubit has parallel north-south qubit couplers ofa same direction—that is each measurement qubit is coupled to asoutherly data qubit. For convenience the couplers in each paired dataand measurement qubit are shown as north-south couplers, however thecouplers could also be south-north (where each measurement qubit iscoupled to a northerly data qubit), east-west (where each measurementqubit is coupled to a westerly data qubit) or west-east couplers (whereeach measurement qubit is coupled to an easterly data qubit).

The first example two dimensional array of qubits 600 includes threenon-overlapping subsets. Each subset includes multiple paired data andneighboring measurement qubits. With reference to FIG. 6, a first subsetincludes all qubits encapsulated by the solid lines, e.g., includingqubit pairs 602, 624 and 622. A second subset includes all qubitsencapsulated by the thick dashed lines, e.g., including qubit pair 604.A third subset includes all qubits encapsulated by the thin dashedlines, e.g., including qubit pair 606. In some cases, as illustrated inqubit array 600, the pairing of data and neighboring measurement qubitsmay not be exhaustive. For example, some qubits at the perimeter of thegrid may not be paired with other qubits.

Each subset includes non-adjacent pairs of data qubits and neighboringmeasurement qubits, where a qubit is said to be adjacent to anotherqubit if it is coupled to the other qubit or is diagonal to the otherqubit. That is, pairs in each subset do not neighbor other pairs in thesubset. Therefore, when entangling operations are performedapproximately in parallel on each pair of data and neighboringmeasurement qubits within a respective subset, each qubit involved in arespective entangling operation is non-adjacent to other qubits involvedin other respective entangling operations. For example, when entanglingoperations are performed in parallel on the pairs included in the subsetrepresented by solid lines, the measurement qubit in the pair 622 canvary its frequency without crossing the resonance of another measurementqubit that is performing a similar frequency trajectory, since themeasurement qubits diagonal to the measurement qubit in the pair 622 aremembers of the other subsets represented by the thick and the thindashed lines. As described above, this configuration reduces theprobability of parasitic occupation qubit leakage.

Each paired data and measurement qubit in the two dimensional array 650has parallel qubit couplers of different directions, that is east-westcouplers (where each measurement qubit is coupled to a westerly dataqubit) or west-east couplers (where each measurement qubit is coupled toan easterly data qubit). In other words, each measurement qubit in thearray 650 may either be coupled to a data qubit via an east-westcoupler, a data qubit via a west-east coupler, or both. Similarly, eachdata qubit in the array 650 may either be coupled to a measurement qubitvia an east-west coupler, a data qubit via a west-east coupler, or both.For convenience, the couplers in each paired data and measurement qubitare shown as east-west and west-east couplers, however the couplerscould also be north-south and south-north couplers.

In some implementations the above described pattern may further berepeated using north-south or south-north couplers, such that allnearest neighbor data qubit and measurement qubit pairs can undergointeractions.

The second example two dimensional array of qubits 650 includes fouroverlapping subsets. Each subset includes multiple paired data andneighboring measurement qubits. With reference to FIG. 6, a first subsetincludes all qubits encapsulated by the solid lines, e.g., includingqubit pair 654. A second subset includes all qubits encapsulated by thethick dashed lines, e.g., qubit pair 658. A third subset includes allqubits encapsulated by the thin dashed lines, e.g., qubit pair 656. Afourth subset includes all qubits encapsulated by the dotted lines,e.g., qubit pair 652. In some cases, as illustrated in qubit array 650,the pairing of data and measurement qubits may be exhaustive, that iseach qubit may be paired with at least one other qubit.

Each subset includes adjacent pairs of data qubits and neighboringmeasurement qubits. For example, either a data qubit in a respectivesubset is diagonal to at least one other data qubit in the subset or ameasurement qubit in a respective subset is diagonal to at least oneother measurement qubit in the subset. Therefore, when entanglingoperations are performed approximately in parallel on each pair of dataand neighboring measurement qubits within a respective subset, eachqubit involved in a respective entangling operation is adjacent(diagonal) to other qubits of a same type involved in other respectiveentangling operations. However, by detuning each measurement qubit inthe subset in parallel, e.g., by maintaining approximately constantdetuning Δf between the measurement qubits in the paired data andmeasurement qubits, each measurement qubit can detuned without crossingthe resonance of another measurement qubit that is performing a similarfrequency trajectory on its corresponding data qubit. As describedabove, this configuration reduces the probability of parasiticoccupation qubit leakage.

FIG. 7 is a plot 700 of an example controlled-Z quantum gate frequencytrajectory 702. The plot 700 shows an example control frequencyamplitude (ΔH_(z)) versus normalized time during application of anadiabatic controlled-Z quantum gate, as described above with referenceto FIG. 5. For example, the control frequency amplitude may representthe amplitude of a control pulse for a controlled-Z quantum gate asgenerated by excitation pulse generator 110 and emitted by excitationdriveline 124 of FIG. 1 above.

The example frequency trajectory 702 may be applied to a paired dataqubit and measurement qubit in order to perform an entangling operation,e.g., a controlled-Z quantum gate. As described above with reference toFIG. 5, in some implementations frequency trajectories appliedapproximately in parallel to respective pairs of data and measurementqubits may include variations in control pulse amplitudes. For example,the values of the control frequency amplitude (ΔH_(z)) may vary, e.g.,by a factor of 100 MHz, to that shown in plot 700.

FIG. 8 is an example plot 800 of the probability of parasitic occupationleakage versus diagonal coupling strength when performing an entanglingoperation on paired data and measurement qubits in parallel, asdescribed above with reference to FIG. 5.

The probability of parasitic occupation qubit leakage during a standardentangling operation on a paired data and measurement qubit, e.g., anentangling operation different to that described in the presentdisclosure, can be estimated using the framework of Landau-Zenertransitions. The framework of Landau-Zener transitions is described, forexample, in “Fast adiabatic qubit gates using only σ_(z) control,” J.Martinis and M. Geller, Phys. Rev. A 90, 022307 (2014), the disclosureof which is incorporated herein by reference in its entirety. Withinthis framework, the probability of occupation leakage can be given bythe equation below:

$P = {{2\left( {1 - {\exp \mspace{9mu} \left( {- \frac{\pi H_{x}^{2}}{\hslash \; {\overset{.}{H}}_{z}}} \right)}} \right)} \approx {2\pi \frac{H_{x}^{2}}{\hslash \; {\overset{.}{H}}_{z}}}}$

In the above equation, H_(x) represents qubit coupling strength andH_(z)={dot over (d)}H_(z)/dt represents a control pulse implementing thestandard entangling operation.

Taking H_(x)=ℏ√{square root over (2)}g_(diag), the rate of frequencychange during the entangling operation frequency trajectory may beestimated as {dot over (f)}=0.5 GHz/10 ns, and the rate of change of thecontrol pulse may be estimated as {dot over (H)}_(z)=ℏf. Inserting thesevalues into the above equation gives P=0.04. This probability ofparasitic occupation qubit leakage is a significant detrimentaloccupation leakage.

The probability P of parasitic occupation leakage during an entanglingoperation as described by this specification can be estimated as:

P = θ_(mr)²/4$\theta_{mr} = {- {\int{\frac{d\theta}{d\tau}{\exp \left( {{- i}\; \omega_{x}\tau} \right)}d{\tau.}}}}$

In the above equation, θ=arctan(H_(x)/H_(z)) is a phase associated witha control pulse implementing the entangling operation, H_(x)=ℏ√{squareroot over (2g)} with g representing the coupling strength between thedata qubit and measurement qubit, H_(z) represents the control pulse,θ_(mr) represents the error angle in the moving and rotating frame, andω_(x)=2H_(x)/ℏ.

Using this framework, the probability of parasitic occupation leakageduring the entangling operation according to the present disclosure isplotted as a function of parasitic coupling strength in plot 800. Inplot 800, the detuning frequency Δf, as described above with referenceto FIG. 5, varies from 500 MHz to 400 MHz, e.g., due to a variation incontrol pulse amplitudes between the qubits, and η=200 MHz. Plot 800shows that the probability of parasitic occupation leakage remains<10⁻¹¹for diagonal coupling strengths g_(diag)/2π (MHz) between 10⁵ and 10⁷.This provides a stark improvement to the probability in settings wherethe energy levels of diagonal qubits cross—an improvement of the orderof 10⁹.

Programming the Hardware: Surface Code Cycle

In some settings, quantum computers can provide a means to efficientlysolve certain problems that may not be efficiently solved using aconventional, classical computer. Example problems include factoringvery large numbers into their primes and searching large, unstructureddata sets. However, physical systems such as systems of ions, spins insemiconductors, and superconducting circuits may not always performsufficiently well to serve directly as computational qubits in a quantumcomputing device.

One approach to building a quantum computing device is based on surfacecodes. Surface codes provide an error-tolerant method for representinginformation in the quantum computing device. Logical qubits areconstructed from collections of physical qubits in such a way that thelogical qubit can perform better than the individual physical qubits.

In some cases surface codes may be operated as stabilizer codes—a methodwhereby stabilizers are measured in order to detect errors as theyarise. By choosing a suitable choice of stabilizer measurements, qubitscan be operated to perform logical operations. Measuring stabilizersover a system of qubits therefore constitutes a fundamental repeatingcycle for the quantum computer, and all higher functions can be builtupon it.

FIG. 9 shows an example quantum circuit 900 to measure a stabilizer fora surface code error detection cycle. The example quantum circuit 900includes a five qubit register. The five qubit register includes ameasurement qubit, represented as |0

, and four data qubits representing the measurement qubits nearestneighbors. In the example quantum circuit the measurement qubit isassumed to be located in a two-dimensional grid, as described withreference to FIG. 1. The four nearest neighboring data qubits thereforecorrespond to a southerly data qubit |S

, westerly data qubit |W

, easterly data qubit |E

and northerly data qubit |N

.

In some cases a quantum circuit may have a smaller qubit register, e.g.,in cases where the measurement qubit has less neighboring data qubits.For example, if the measurement qubit is at a corner of thetwo-dimensional grid, the measurement qubit may only have twoneighboring data qubits. In this example, a corresponding quantumcircuit may have a three qubit register.

The example quantum circuit 900 shows the sequence of quantum logicgates needed to perform the surface code error detection cycle 1000described herein with reference to FIG. 10. As described herein withreference to FIG. 10, the example quantum circuit 900 includes a firstHadamard gate 952 that is applied to the measurement qubit |0

. Subsequently, a first entangling operation 956 is performed on themeasurement qubit register |0

and the southerly data qubit register |S

. A second Hadamard gate 958 is subsequently applied to the westerlydata qubit register |W

. A second entangling operation 960 is then applied to the measurementqubit register |0

and the westerly data qubit register |W

.

The example quantum circuit 900 includes a third and fourth Hadamardgate 972 and 974. The Hadamard gates 972 and 974 are sequentiallyapplied to the westerly qubit |W

and easterly qubit |E

, respectively. When example quantum circuit 900 is applied to a systemof measurement qubits and data qubits, as described above with referenceto FIG. 1, a Hadamard gate applied to a westerly data qubit (after anentangling operation between a first measurement qubit and the westerlydata qubit) is cancelled by a Hadamard gate applied to an easterly dataqubit (before an entangling operation between a second measurement qubitand the easterly data qubit).

A third entangling operation 962 is applied to the measurement qubitregister |0

and the easterly data qubit register |E

. A fifth Hadamard gate 964 is subsequently applied to the easterly dataqubit register |E

. A fourth entangling operation 966 is applied to the measurement qubitregister |0

and the northerly data qubit register |N

. A sixth Hadamard gate is applied to the measurement qubit register |0

, followed by a measurement operation 970.

The entangling operations 956, 960, 962, and 966 may includecontrolled-Z quantum logic gates. When Hadamard quantum logic gates areapplied before and after a controlled-Z quantum logic gate, e.g.,Hadamard quantum logic gates 958 and 972 or 974 and 964, the three gatestogether (Hadamard, controlled-Z, Hadamard) operate as a controlled-Xquantum logic gate. Therefore, taken collectively, the entanglingoperations depicted in FIG. 9 may represent an application of theoperator ZXXZ (controlled-Z, controlled-X, controlled-X, controlled-Z)if the measurement qubit is in the |1

state.

FIG. 10 is a flow diagram of an example process 1000 for performing asurface code error detection cycle on multiple quantum circuits e.g.quantum circuits shown in FIG. 9. For convenience, the process 1000 willbe described as being performed by a system of one or more quantum orclassical computers located in one or more locations. For example, theprocess 1000 can be implemented using the qubit controller 104 of system100 described above with reference to FIG. 1. In some implementations,the process 1000 may be performed in conjunction with the techniquesdescribed above with reference to FIGS. 2 to 8.

The example process 1000 is described as being performed by the systemon multiple data qubits and multiple measurement qubits arranged as atwo dimensional grid, e.g., grid 112 of FIG. 1, each data qubit of themultiple data qubits being coupled to neighboring measurement qubitsthrough respective qubit couplers, as described above with reference toFIG. 1.

The system initializes the multiple measurement qubits (step 1002). Forexample, as illustrated in the example quantum circuit 900 of FIG. 9,initializing the multiple measurement qubits may include preparing themeasurement qubits in the |0

computational basis state.

The system applies Hadamard quantum logic gates to the initializedmeasurement qubits (step 1004). By initializing the measurement qubitsin the |0

computational basis state and applying Hadamard quantum logic gates tothe initialized measurement qubits, the measurement qubits are put it ina 50/50 superposition state of |0

and |1

. Application of a Hadamard quantum logic gate 952 to an initializedmeasurement qubit is illustrated above with reference to FIG. 9.

The system performs multiple entangling operations on a first set ofpaired measurement and data qubits (step 1006). For example, theentangling operations may include controlled-Z quantum logic gates.Application of a controlled-Z quantum logic gate to a paired measurementand data qubit includes applying a Z operator to the data qubit if themeasurement qubit is in the state |1

.

Each pair in the first set of paired measurement and data qubitsincludes a measurement qubit coupled to a neighboring data qubit in afirst direction. For example, each pair may include a measurement qubitcoupled via a respective qubit coupler to a neighboring data qubit thatis below the measurement qubit, e.g., in a southerly direction. Examplepairs of measurement qubits coupled to respective neighboring dataqubits in a southerly direction are illustrated and described hereinwith reference to two dimensional qubit grid 600 of FIG. 6. Anapplication of an entangling operation 956 to a measurement qubit pairedwith a southerly data qubit is illustrated with reference to FIG. 9.

In some implementations, performing multiple entangling operations onthe first set of paired measurement and data qubits includes separatingthe paired measurement and data qubits into multiple subsets of pairedqubits, the multiple subsets including non-overlapping and non-adjacentpairs. In these implementations, non-adjacent is understood to includediagonally non-adjacent pairs. Example multiple subsets of paired qubitsare illustrated and described above with reference to two dimensionalqubit grid 600 of FIG. 6. As illustrated in two dimensional qubit grid600 of FIG. 6, in some implementations the multiple subsets may includethree subsets 602, 604, and 606.

The system may then perform entangling operations on the pairs of qubitsin each of the multiple subsets in parallel. For example, as describedwith reference to FIG. 5, performing entangling operations on pairs ofqubits in each of the multiple subsets in parallel may include detuningeach measurement qubit in each subset in parallel.

The system applies Hadamard quantum logic gates to the multiple dataqubits in the second direction (step 1008). For example, the system mayapply Hadamard quantum logic gates to multiple data qubits in a westerlydirection from the measurement qubits. Application of a Hadamard quantumlogic gate 958 to westerly data qubits is illustrated with reference toFIG. 9.

The system performs multiple operations on a second set of pairedmeasurement and data qubits (step 1010). The operations may includecontrolled-Z quantum logic gates and Hadamard quantum logic gates. Forexample, the system may perform controlled-Z quantum logic gates onmeasurement qubits paired with data qubits in a second direction,followed by performing Hadamard quantum logic gates on the data qubitsin the second direction. The system may then perform Hadamard quantumlogic gates on the data qubits in a third direction, followed byperforming controlled-Z quantum gates on the measurement qubits pairedwith the data qubits in the third direction.

Each pair in the second set of paired measurement and data qubitsincludes a measurement qubit coupled to a neighboring data qubit in asecond or third direction, the second and third direction beingperpendicular to the first direction, and the second direction beingopposite to the third direction. For example, each pair may include ameasurement qubit coupled via a respective qubit coupler to aneighboring data qubit that is to the right or to the left of themeasurement qubit, i.e., in an easterly or westerly direction. Sincewesterly and easterly entangling operations commute, the system mayperform a mix of westerly and easterly entangling operations.

Example pairs of measurement qubits coupled to respective neighboringdata qubits in an easterly and westerly direction are illustrated anddescribed above with reference to two dimensional qubit grid 650 of FIG.6. An application of entangling operations 960 and 962 applied to ameasurement qubit paired with a westerly data qubit and an easterly dataqubit, respectively, is illustrated with reference to FIG. 9.

In some implementations, performing multiple entangling operations onthe second set of paired measurement and data qubits includes separatingthe paired measurement and data qubits into multiple subsets of pairedqubits, the multiple subsets including overlapping and adjacent pairs.In these implementations, adjacent is understood to include diagonallyadjacent pairs. Example multiple subsets of such paired qubits areillustrated and described above with reference to two dimensional qubitgrid 650 of FIG. 6. As illustrated in two dimensional qubit grid 650 ofFIG. 6, in some implementations the multiple subsets may include foursubsets 652, 654, 656 and 658.

The system may then perform the operations on pairs of qubits in each ofthe multiple subsets in parallel. For example, as described withreference to FIG. 5, performing entangling operations on pairs of qubitsin each of the multiple subsets in parallel may include detuning eachmeasurement qubit in each subset in parallel.

The system applies Hadamard quantum logic gates to the multiple dataqubits in the third direction (step 1012). For example, the system mayapply Hadamard quantum logic gates to multiple data qubits in aneasterly direction from the measurement qubits. Application of aHadamard quantum logic gate 964 to easterly data qubits is illustratedwith reference to FIG. 9. As described with reference to FIG. 9, whenHadamard quantum logic gates are applied before and after a controlled-Zquantum logic gate, e.g., as described with reference to steps 1010 and1012, the three gates together act as a controlled-X quantum logic gate.

The system performs multiple entangling operations to a third set ofpaired measurement and data qubits (step 1014). As described above, theentangling operations may include controlled-Z quantum logic gates. Eachpair in the third set of paired measurement and data qubits includes ameasurement qubit coupled to a neighboring data qubit in a fourthdirection, the fourth direction being opposite to the first direction.For example, each pair may include a measurement qubit coupled via arespective qubit coupler to a neighboring data qubit that is above themeasurement qubit, i.e., in a northerly direction. Example pairs ofmeasurement qubits coupled to respective neighboring data qubits in anortherly direction can result from a straightforward modification ofthe two dimensional qubit grid 600 of FIG. 6. An application of anentangling operation 966 to a measurement qubit paired with a northerlydata qubit is illustrated with reference to FIG. 9.

In some implementations, performing multiple entangling operations onthe third set of paired measurement and data qubits includes separatingthe paired measurement and data qubits into multiple subsets of pairedqubits, the multiple subsets including non-overlapping and non-adjacentpairs. In these implementations, non-adjacent is understood to includediagonally non-adjacent pairs. In some implementations the multiplesubsets may include three subsets.

The system may then perform entangling operations on the pairs of qubitsin each of the multiple subsets in parallel. For example, as describedwith reference to FIG. 5, performing entangling operations on pairs ofqubits in each of the multiple subsets in parallel may include detuningeach measurement qubit in each subset in parallel.

The system applies Hadamard quantum logic gates to the multiplemeasurement qubits (step 1016). Application of a Hadamard quantum logicgate 968 to a measurement qubit is illustrated with reference to FIG. 9.

The system measures the multiple measurement qubits to detect errors(step 1018). An example measurement operation 970 is illustrated withreference to FIG. 9.

As described above, performing multiple entangling operations on thefirst set of paired measurement and data qubits requires threesequential applications of arrays of entangling operations—oneapplication for each subset. If this scheme were applied individuallyfor all four directions of nearest neighbor interaction, e.g., north,south, east and west, the complete surface code error detection cycle1000 would require 12 applications of entangling operations. However, bydetuning geometrically diagonal measurement qubits in parallel, e.g.,following the techniques described above with reference to FIGS. 5 and6, denser patterns of entangling operations enables all interactionsperpendicular to the first direction, e.g., east and west, to becompleted in just four layers of CZ gate—one application for eachsubset—resulting in a total of just ten applications of entanglingoperations.

Optionally, the system may further perform leakage removal. For example,the system may perform leakage removal concurrently with eachmeasurement qubit's final entangling operation, e.g., concurrently withstep 1014 described above. For example, the system may swap themeasurement and data qubits such that each type of qubit isalternatively reset. This may be achieved by applying a controlled-Zplus swap quantum logic gate that interacts and transfers information inthe computational basis states |0

and |1

but does not transfer information in states |2

and higher.

In some implementations a subsequent surface code error detection cyclemay be performed in an inverted order to the cycle described in steps1002-1018 above. For example, instead of performing asouth—west/east—north detection cycle, as described above, the systemmay perform a north—west/east—south detection cycle. That is, the systemmay initialize the multiple measurement qubits, apply Hadamard quantumlogic gates to the initialized measurement qubits, perform entanglingoperations on the third subset of paired data and measurement qubits inparallel, apply Hadamard quantum logic gates to the multiple dataqubits, perform entangling operations on the second subset of paireddata and measurement qubits in parallel; apply Hadamard quantum logicgates to the multiple data qubits, perform entangling operations on thefirst subset of paired data and measurement qubits in parallel, applyHadamard quantum logic gates to the multiple measurement qubits andmeasure the multiple measurement qubits to detect errors. Performing thesubsequent surface code error detection cycle in this order can ensurethat data remains local, e.g., that information read out from eachpaired measurement qubit corresponds only to a respective data qubit.

FIG. 11 shows an example implementation of the surface code 1100. Theexample implementation 1100 shows a two dimensional array of qubits, asdescribed above with reference to FIG. 1. Each of the qubits in the twodimensional array of qubits is represented as an open circle, e.g.,qubit 1104, or a filled-in circle, e.g., 1106. In some implementationsthe open circles represent data qubits, as described above withreference to FIG. 1. In these implementations, the filled-in circlesrepresent measurement qubits, as described above with reference toFIG. 1. For clarity, the two dimensional array of qubits includes 5×5qubits, however in some cases implementations of the surface code mayinclude a smaller or a larger number of qubits.

As described above with reference to FIG. 1, the qubits interact witheach other through multiple nearest neighbor qubit couplers which, forconvenience, as not shown in example implementation 1100. Therefore,away from the array boundary, each data qubit contacts four measurementqubits, and each measurement qubit contacts four data qubits. Themeasurement qubits therefore perform four measurements. On the arrayboundary, the measurement qubits contact three data qubits and performthree measurements, and the data qubits contact either two or threemeasurement qubits.

The example implementation 1100 includes multiple uniform stabilizers,e.g., stabilizer 1102. The stabilizers are used to preserve the quantumstate of the array of qubits. Generally, by repeatedly measuring aquantum system using a complete set of commuting stabilizers, thequantum system is forced into a simultaneous and unique eigenstate ofall the stabilizers. The stabilizers can be measured without perturbingthe system. When the measurement outcomes change, this corresponds toone or more qubit errors, and the quantum state is projected by themeasurements onto a different stabilizer eigenstate. Surface codestabilizers are described, for example, in “Surface codes: Towardspractical large-scale quantum computation,” A. Fowler et al, Phys. Rev.A 86, 032324 (2012), the disclosure of which is incorporated herein byreference in its entirety.

Each stabilizer in the example implementation 1102 includes a product of{circumflex over (Z)}={circumflex over (σ)}_(z) and {circumflex over(X)}={circumflex over (σ)}_(x) operators. For example, stabilizer 1102may be represented as {circumflex over (Z)}_(n){circumflex over(X)}_(w){circumflex over (X)}_(e){circumflex over (Z)}_(s) where theindices n, w, e and s represent the directions north, west, east, andsouth with respect to the data qubit on which the stabilizer operateson. By combining {circumflex over (Z)} and {circumflex over (X)}operators in this manner, opposite interactions, e.g., east-westinteractions, may be performed simultaneously—where simultaneously isunderstood to mean simultaneously to the extent that the hardware usedto implement the surface code allows—since the east-west operatorscommute.

Embodiments of the digital and/or quantum subject matter and the digitalfunctional operations and quantum operations described in thisspecification can be implemented in digital electronic circuitry,suitable quantum circuitry or, more generally, quantum computationalsystems, in tangibly-embodied digital and/or quantum computer softwareor firmware, in digital and/or quantum computer hardware, including thestructures disclosed in this specification and their structuralequivalents, or in combinations of one or more of them. The term“quantum computational systems” may include, but is not limited to,quantum computers, quantum information processing systems, quantumcryptography systems, or quantum simulators.

Embodiments of the digital and/or quantum subject matter described inthis specification can be implemented as one or more digital and/orquantum computer programs, i.e., one or more modules of digital and/orquantum computer program instructions encoded on a tangiblenon-transitory storage medium for execution by, or to control theoperation of, data processing apparatus. The digital and/or quantumcomputer storage medium can be a machine-readable storage device, amachine-readable storage substrate, a random or serial access memorydevice, one or more qubits, or a combination of one or more of them.Alternatively or in addition, the program instructions can be encoded onan artificially-generated propagated signal that is capable of encodingdigital and/or quantum information, e.g., a machine-generatedelectrical, optical, or electromagnetic signal, that is generated toencode digital and/or quantum information for transmission to suitablereceiver apparatus for execution by a data processing apparatus.

The terms quantum information and quantum data refer to information ordata that is carried by, held or stored in quantum systems, where thesmallest non-trivial system is a qubit, i.e., a system that defines theunit of quantum information. It is understood that the term “qubit”encompasses all quantum systems that may be suitably approximated as atwo-level system in the corresponding context. Such quantum systems mayinclude multi-level systems, e.g., with two or more levels. By way ofexample, such systems can include atoms, electrons, photons, ions orsuperconducting qubits. In many implementations the computational basisstates are identified with the ground and first excited states, howeverit is understood that other setups where the computational states areidentified with higher level excited states are possible.

The term “data processing apparatus” refers to digital and/or quantumdata processing hardware and encompasses all kinds of apparatus,devices, and machines for processing digital and/or quantum data,including by way of example a programmable digital processor, aprogrammable quantum processor, a digital computer, a quantum computer,multiple digital and quantum processors or computers, and combinationsthereof. The apparatus can also be, or further include, special purposelogic circuitry, e.g., an FPGA (field programmable gate array), an ASIC(application-specific integrated circuit), or a quantum simulator, i.e.,a quantum data processing apparatus that is designed to simulate orproduce information about a specific quantum system. In particular, aquantum simulator is a special purpose quantum computer that does nothave the capability to perform universal quantum computation. Theapparatus can optionally include, in addition to hardware, code thatcreates an execution environment for digital and/or quantum computerprograms, e.g., code that constitutes processor firmware, a protocolstack, a database management system, an operating system, or acombination of one or more of them.

A digital computer program, which may also be referred to or describedas a program, software, a software application, a module, a softwaremodule, a script, or code, can be written in any form of programminglanguage, including compiled or interpreted languages, or declarative orprocedural languages, and it can be deployed in any form, including as astand-alone program or as a module, component, subroutine, or other unitsuitable for use in a digital computing environment. A quantum computerprogram, which may also be referred to or described as a program,software, a software application, a module, a software module, a script,or code, can be written in any form of programming language, includingcompiled or interpreted languages, or declarative or procedurallanguages, and translated into a suitable quantum programming language,or can be written in a quantum programming language, e.g., QCL orQuipper.

A digital and/or quantum computer program may, but need not, correspondto a file in a file system. A program can be stored in a portion of afile that holds other programs or data, e.g., one or more scripts storedin a markup language document, in a single file dedicated to the programin question, or in multiple coordinated files, e.g., files that storeone or more modules, sub-programs, or portions of code. A digital and/orquantum computer program can be deployed to be executed on one digitalor one quantum computer or on multiple digital and/or quantum computersthat are located at one site or distributed across multiple sites andinterconnected by a digital and/or quantum data communication network. Aquantum data communication network is understood to be a network thatmay transmit quantum data using quantum systems, e.g. qubits. Generally,a digital data communication network cannot transmit quantum data,however a quantum data communication network may transmit both quantumdata and digital data.

The processes and logic flows described in this specification can beperformed by one or more programmable digital and/or quantum computers,operating with one or more digital and/or quantum processors, asappropriate, executing one or more digital and/or quantum computerprograms to perform functions by operating on input digital and quantumdata and generating output. The processes and logic flows can also beperformed by, and apparatus can also be implemented as, special purposelogic circuitry, e.g., an FPGA or an ASIC, or a quantum simulator, or bya combination of special purpose logic circuitry or quantum simulatorsand one or more programmed digital and/or quantum computers.

For a system of one or more digital and/or quantum computers to be“configured to” perform particular operations or actions means that thesystem has installed on it software, firmware, hardware, or acombination of them that in operation cause the system to perform theoperations or actions. For one or more digital and/or quantum computerprograms to be configured to perform particular operations or actionsmeans that the one or more programs include instructions that, whenexecuted by digital and/or quantum data processing apparatus, cause theapparatus to perform the operations or actions. A quantum computer mayreceive instructions from a digital computer that, when executed by thequantum computing apparatus, cause the apparatus to perform theoperations or actions.

Digital and/or quantum computers suitable for the execution of a digitaland/or quantum computer program can be based on general or specialpurpose digital and/or quantum processors or both, or any other kind ofcentral digital and/or quantum processing unit. Generally, a centraldigital and/or quantum processing unit will receive instructions anddigital and/or quantum data from a read-only memory, a random accessmemory, or quantum systems suitable for transmitting quantum data, e.g.photons, or combinations thereof.

The essential elements of a digital and/or quantum computer are acentral processing unit for performing or executing instructions and oneor more memory devices for storing instructions and digital and/orquantum data. The central processing unit and the memory can besupplemented by, or incorporated in, special purpose logic circuitry orquantum simulators. Generally, a digital and/or quantum computer willalso include, or be operatively coupled to receive digital and/orquantum data from or transfer digital and/or quantum data to, or both,one or more mass storage devices for storing digital and/or quantumdata, e.g., magnetic, magneto-optical disks, optical disks, or quantumsystems suitable for storing quantum information. However, a digitaland/or quantum computer need not have such devices.

Digital and/or quantum computer-readable media suitable for storingdigital and/or quantum computer program instructions and digital and/orquantum data include all forms of non-volatile digital and/or quantummemory, media and memory devices, including by way of examplesemiconductor memory devices, e.g., EPROM, EEPROM, and flash memorydevices; magnetic disks, e.g., internal hard disks or removable disks;magneto-optical disks; CD-ROM and DVD-ROM disks; and quantum systems,e.g., trapped atoms or electrons. It is understood that quantum memoriesare devices that can store quantum data for a long time with highfidelity and efficiency, e.g., light-matter interfaces where light isused for transmission and matter for storing and preserving the quantumfeatures of quantum data such as superposition or quantum coherence.

Control of the various systems described in this specification, orportions of them, can be implemented in a digital and/or quantumcomputer program product that includes instructions that are stored onone or more non-transitory machine-readable storage media, and that areexecutable on one or more digital and/or quantum processing devices. Thesystems described in this specification, or portions of them, can eachbe implemented as an apparatus, method, or system that may include oneor more digital and/or quantum processing devices and memory to storeexecutable instructions to perform the operations described in thisspecification.

While this specification contains many specific implementation details,these should not be construed as limitations on the scope of what may beclaimed, but rather as descriptions of features that may be specific toparticular embodiments. Certain features that are described in thisspecification in the context of separate embodiments can also beimplemented in combination in a single embodiment. Conversely, variousfeatures that are described in the context of a single embodiment canalso be implemented in multiple embodiments separately or in anysuitable sub-combination. Moreover, although features may be describedabove as acting in certain combinations and even initially claimed assuch, one or more features from a claimed combination can in some casesbe excised from the combination, and the claimed combination may bedirected to a sub-combination or variation of a sub-combination.

Similarly, while operations are depicted in the drawings in a particularorder, this should not be understood as requiring that such operationsbe performed in the particular order shown or in sequential order, orthat all illustrated operations be performed, to achieve desirableresults. In certain circumstances, multitasking and parallel processingmay be advantageous. Moreover, the separation of various system modulesand components in the embodiments described above should not beunderstood as requiring such separation in all embodiments, and itshould be understood that the described program components and systemscan generally be integrated together in a single software product orpackaged into multiple software products.

Particular embodiments of the subject matter have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results. As one example, the processesdepicted in the accompanying figures do not necessarily require theparticular order shown, or sequential order, to achieve desirableresults. In some cases, multitasking and parallel processing may beadvantageous.

1. A method for performing a surface code error detection cycle, themethod comprising: initializing multiple measurement qubits from asystem comprising a plurality of data qubits and a plurality ofmeasurement qubits arranged as a two-dimensional grid, wherein each dataqubit of the plurality of data qubits within the grid is coupled toneighboring measurement qubits through respective qubit couplers;applying Hadamard quantum logic gates to the initialized measurementqubits; and performing multiple entangling operations on a first set ofpaired measurement and data qubits, wherein each pair in the first setof paired measurement and data qubits comprises a measurement qubitcoupled to a neighboring data qubit in a first direction; performingmultiple entangling operations to a second set of paired measurement anddata qubits, wherein each pair in the second set of paired measurementand data qubits comprises a measurement qubit coupled to a neighboringdata qubit in a second or third direction, the second and thirddirection being perpendicular to the first direction, the seconddirection being opposite to the third direction; performing multipleentangling operations to a third set of paired measurement and dataqubits, wherein each pair in the third set of paired measurement anddata qubits comprises a measurement qubit coupled to a neighboring dataqubit in a fourth direction, the fourth direction being opposite to thefirst direction; applying Hadamard quantum logic gates to the multiplemeasurement qubits; and measuring the multiple measurement qubits todetect errors.
 2. The method of claim 1, further comprising applyingHadamard quantum logic gates to data qubits in the second set of pairedmeasurement and data qubits that are paired with measurement qubits inthe second direction.
 3. The method of claim 1, further comprising:applying Hadamard quantum logic gates to data qubits in the second setof paired measurement and data qubits that are paired with measurementqubits in the second direction; and applying Hadamard quantum logicgates to data qubits in the second set of paired measurement and dataqubits that are paired with measurement qubits in the third direction.4. The method of claim 1, further comprising: applying Hadamard quantumlogic gates to data qubits in the second set of paired measurement anddata qubits that are paired with measurement qubits in the thirddirection.
 5. The method of claim 1, wherein the entangling operationscomprise controlled-Z quantum logic gates.
 6. The method of claim 1,wherein performing multiple entangling operations on the first set ofpaired measurement and data qubits comprises: separating the pairs intomultiple subsets of paired qubits, the multiple subsets comprising nonoverlapping and non-adjacent pairs.
 7. The method of claim 6, whereinthe multiple subsets comprise three subsets.
 8. The method of claim 6,wherein performing multiple entangling operations to the first set ofpaired measurement and data qubits comprises performing entanglingoperations on pairs of qubits in each of the multiple subsets inparallel.
 9. The method of claim 8, wherein performing entanglingoperations on pairs of qubits in each of the multiple subsets inparallel comprises detuning each measurement qubit in each subset inparallel.
 10. The method of claim 1, wherein performing multipleentangling operations on the second set of paired measurement and dataqubits comprises: separating the pairs into multiple subsets of pairedqubits, the multiple subsets comprising overlapping and diagonallyadjacent pairs.
 11. The method of claim 10, wherein the multiple subsetscomprise four subsets.
 12. The method of claim 11, wherein performingmultiple entangling operations on the second set of paired measurementand data qubits comprises performing entangling operations on pairs ofqubits in each of the multiple subsets in parallel.
 13. The method ofclaim 12, wherein performing entangling operations on pairs of qubits ineach of the multiple subsets in parallel comprises detuning eachmeasurement qubit in the each subset in parallel.
 14. The method ofclaim 1, wherein performing multiple entangling operations on the thirdset of paired measurement and data qubits comprises: separating thepairs into multiple subsets of paired qubits, the multiple subsetscomprising non overlapping and non-adjacent pairs.
 15. The method ofclaim 14, wherein the multiple subsets comprise three subsets.
 16. Themethod of claim 15, wherein performing multiple entangling operations onthe third set of paired measurement and data qubits comprises performingentangling operations on pairs of qubits in each of the multiple subsetsin parallel.
 17. The method of claim 16, wherein performing entanglingoperations on pairs of qubits in each of the multiple subsets inparallel comprises detuning each measurement qubit in the each subset inparallel.
 18. The method of claim 1, further comprising performing aleakage removal process on the measurement and data qubits.
 19. Themethod of claim 18, wherein performing leakage removal comprisesswapping measurement and data qubits.
 20. The method of claim 19,wherein swapping measurement and data qubits comprises applying acontrolled-Z plus swap quantum gate.
 21. The method of claim 1, furthercomprising performing a subsequent surface code error detection cycle,comprising: initializing the multiple measurement qubits; applyingHadamard quantum logic gates to the initialized measurement qubits;performing entangling operations on the third subset of paired data andmeasurement qubits in parallel; performing entangling operations on thesecond subset of paired data and measurement qubits in parallel;performing entangling operations on the first subset of paired data andmeasurement qubits in parallel; applying Hadamard quantum logic gates tothe multiple measurement qubits; and measuring the multiple measurementqubits to detect errors.
 22. An apparatus comprising: a plurality ofdata qubits; a plurality of measurement qubits; a plurality of qubitcouplers defining nearest neighbor interactions between the data qubitsand measurement qubits, wherein the multiple data qubits and measurementqubits are arranged as a two dimensional grid, each data qubit of themultiple data qubits is coupled to multiple measurement qubits throughrespective qubit couplers; and a qubit controller module configured tooperate the multiple data qubits and multiple measurement qubits,wherein the qubit controller is configured to perform a surface codeerror detection cycle on the multiple data qubits and multiplemeasurement qubits, comprising: initializing multiple measurement qubitsfrom a system comprising a plurality of data qubits and a plurality ofmeasurement qubits arranged as a two-dimensional grid, wherein each dataqubit of the plurality of data qubits within the grid is coupled toneighboring measurement qubits through respective qubit couplers;applying Hadamard quantum logic gates to the initialized measurementqubits; and performing multiple entangling operations on a first set ofpaired measurement and data qubits, wherein each pair in the first setof paired measurement and data qubits comprises a measurement qubitcoupled to a neighboring data qubit in a first direction; performingmultiple entangling operations to a second set of paired measurement anddata qubits, wherein each pair in the second set of paired measurementand data qubits comprises a measurement qubit coupled to a neighboringdata qubit in a second or third direction, the second and thirddirection being perpendicular to the first direction, the seconddirection being opposite to the third direction; performing multipleentangling operations to a third set of paired measurement and dataqubits, wherein each pair in the third set of paired measurement anddata qubits comprises a measurement qubit coupled to a neighboring dataqubit in a fourth direction, the fourth direction being opposite to thefirst direction; applying Hadamard quantum logic gates to the multiplemeasurement qubits; and measuring the multiple measurement qubits todetect errors.
 23. The apparatus of claim 22, wherein the qubitcontroller is further configured to apply Hadamard quantum logic gatesto data qubits in the second set of paired measurement and data qubitsthat are paired with measurement qubits in the second direction.
 24. Theapparatus of claim 22, wherein the qubit controller is furtherconfigured to: apply Hadamard quantum logic gates to data qubits in thesecond set of paired measurement and data qubits that are paired withmeasurement qubits in the second direction; and apply Hadamard quantumlogic gates to data qubits in the second set of paired measurement anddata qubits that are paired with measurement qubits in the thirddirection.
 25. The apparatus of claim 22, wherein the qubit controlleris further configured to apply Hadamard quantum logic gates to dataqubits in the second set of paired measurement and data qubits that arepaired with measurement qubits in the third direction.
 26. The apparatusof claim 22, wherein the entangling operations comprise controlled-Zquantum logic gates.
 27. The apparatus of claim 22, wherein performingmultiple entangling operations on the first set of paired measurementand data qubits comprises: separating the pairs into multiple subsets ofpaired qubits, the multiple subsets comprising non overlapping andnon-adjacent pairs.
 28. The apparatus of claim 27, wherein the multiplesubsets comprise three subsets.
 29. The apparatus of claim 27, whereinperforming multiple entangling operations to the first set of pairedmeasurement and data qubits comprises performing entangling operationson pairs of qubits in each of the multiple subsets in parallel.
 30. Theapparatus of claim 29, wherein performing entangling operations on pairsof qubits in each of the multiple subsets in parallel comprises detuningeach measurement qubit in each subset in parallel.
 31. The apparatus ofclaim 22, wherein performing multiple entangling operations on thesecond set of paired measurement and data qubits comprises: separatingthe pairs into multiple subsets of paired qubits, the multiple subsetscomprising overlapping and diagonally adjacent pairs.
 32. The apparatusof claim 31, wherein the multiple subsets comprise four subsets.
 33. Theapparatus of claim 32, wherein performing multiple entangling operationson the second set of paired measurement and data qubits comprisesperforming entangling operations on pairs of qubits in each of themultiple subsets in parallel.
 34. The apparatus of claim 33, whereinperforming entangling operations on pairs of qubits in each of themultiple subsets in parallel comprises detuning each measurement qubitin the each subset in parallel.
 35. The apparatus of claim 22, whereinperforming multiple entangling operations on the third set of pairedmeasurement and data qubits comprises: separating the pairs intomultiple subsets of paired qubits, the multiple subsets comprising nonoverlapping and non-adjacent pairs.
 36. The apparatus of claim 35,wherein the multiple subsets comprise three subsets.
 37. The apparatusof claim 36, wherein performing multiple entangling operations on thethird set of paired measurement and data qubits comprises performingentangling operations on pairs of qubits in each of the multiple subsetsin parallel.
 38. The apparatus of claim 37, wherein performingentangling operations on pairs of qubits in each of the multiple subsetsin parallel comprises detuning each measurement qubit in the each subsetin parallel.
 39. The apparatus of claim 22, wherein the qubit controlleris further configured to perform a leakage removal process on themeasurement and data qubits.
 40. The apparatus of claim 39, whereinperforming leakage removal comprises swapping measurement and dataqubits.
 41. The apparatus of claim 40, wherein swapping measurement anddata qubits comprises applying a controlled-Z plus swap quantum gate.42. The apparatus of claim 22, wherein the qubit controller is furtherconfigured to perform a subsequent surface code error detection cycle,comprising: initializing the multiple measurement qubits; applyingHadamard quantum logic gates to the initialized measurement qubits;performing entangling operations on the third subset of paired data andmeasurement qubits in parallel; performing entangling operations on thesecond subset of paired data and measurement qubits in parallel;performing entangling operations on the first subset of paired data andmeasurement qubits in parallel; applying Hadamard quantum logic gates tothe multiple measurement qubits; and measuring the multiple measurementqubits to detect errors.
 43. The apparatus of claim 22, wherein theplurality of data qubits comprise Xmon qubits.
 44. The apparatus ofclaim 22, wherein the plurality of measurement qubits comprise Xmonqubits.